Page 244 - Determinants and Their Applications in Mathematical Physics

5.8 Some Applications of Algebraic Computing 229

H 2 H 4 h 2 h 4
H 5 h 5
H 1 H 3 = h 1 h 3
,
H 4 h 4
1 H 2 H 3 1 h 2 h 3

h 1 h 3 h 4 h 5

1

H 3 h 2 h 3 h 4

. (5.8.2)
H 5
=
H 1 H 3 h 1 h 2

h 3
1 h 1

5.8.3 Hankel Determinants with Hankel Elements
Let
A n = |φ r+m | n , 0 ≤ m ≤ 2n − 2, (5.8.3)
which is an Hankelian (or a Turanian).
Let
B r = A 2

= φ r φ r+1 . (5.8.4)
φ r+1 φ r+2

Then B r , B r+1 , and B r+2 are each Hankelians of order 2 and are each
minors of A 3 :
(3)
B r = A ,
33
(3) (3)
B r+1 = A = A ,
31 13
(3)
B r+2 = A 11 . (5.8.5)
Hence, applying the Jacobi identity (Section 3.6),
(3) (3)
B r+2 11
A A
B r+1 = (3) 13
B r+1 A A
(3)
31 33
B r
(3)
= A 3 A
13,13
= φ 2 A 3 . (5.8.6)
Now redefine B r . Let B r = A 3 . Then, B r , B r+1 ,...,B r+4 are each second
minors of A 5 :
(5)
B r = A ,
45,45
(5) (5)
B r+1 = −A = −A ,
15,45 45,15
(5) (5) (5)
B r+2 = A = A = A ,
12,45 15,15 45,12
(5) (5)
B r+3 = −A 12,15 = −A 15,12 ,
(5)
B r+4 = A . (5.8.7)
12,12